This is joint work with Sheel Ganatra and Vivek Shende. We setup Floer theory for a certain class of Liouville manifolds with boundary which we call "Liouville sectors". Liouville manifolds often have nontrivial covers by Liouville sectors, so this opens up the possibility of obtaining a sheaf-theoretic description of Floer theory on certain Liouville manifolds. In particular, we obtain a local version of Abouzaid's generation criterion, which we verify for Weinstein manifolds whose core has arboreal singularities in the sense of Nadler. We hope to apply similar reasoning to address the conjecture of Kontsevich that the Fukaya category of a Weinstein manifold is isomorphic to a suitable category of sheaves on its core (the core of a Weinstein manifold is the union of stable manifolds of the Liouville flow).